Arithmetic and Dynamical Degrees on Abelian Varieties

Joseph H. Silverman

doi:10.5802/jtnb.973

Joseph H. Silverman ¹

¹ Mathematics Department, Box 1917 Brown University, Providence, RI 02912, USA

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 151-167.

Résumé
Abstract

Soit $φ : X ⤏ X$ une application rationnelle dominante d’une variété lisse et soit $x \in X$ , tous deux définis sur $\bar{ℚ}$ . Le degré dynamique $δ (φ)$ mesure la complexité géométrique des itérations de $φ$ , tandis que le degré arithmétique $α (φ, x)$ mesure la complexité arithmétique de la $φ$ -orbite de $x$ . Il est connu que $α (φ, x) \leq δ (φ)$ , et il est conjecturé que si la $φ$ -orbite de $x$ est Zariski dense dans $X$ , alors $α (φ, x) = δ (φ)$ . Dans cette note, nous prouvons cette conjecture dans le cas où $X$ est une variété abélienne, étendant des travaux antérieurs où la conjecture a été prouvée pour les isogénies.

Let $φ : X ⤏ X$ be a dominant rational map of a smooth variety and let $x \in X$ , all defined over $\bar{ℚ}$ . The dynamical degree $δ (φ)$ measures the geometric complexity of the iterates of $φ$ , and the arithmetic degree $α (φ, x)$ measures the arithmetic complexity of the forward $φ$ -orbit of $x$ . It is known that $α (φ, x) \leq δ (φ)$ , and it is conjectured that if the $φ$ -orbit of $x$ is Zariski dense in $X$ , then $α (φ, x) = δ (φ)$ , i.e. arithmetic complexity equals geometric complexity. In this note we prove this conjecture in the case that $X$ is an abelian variety, extending earlier work in which the conjecture was proven for isogenies.

Reçu le : 2015-03-01
Révisé le : 2015-06-30
Accepté le : 2015-07-06
Publié le : 2017-01-26

DOI : 10.5802/jtnb.973

Classification : 37P30, 11G10, 11G50, 37P15
Mots clés : dynamical degree, arithmetic degree, abelian variety

Affiliations des auteurs :

Joseph H. Silverman ¹

¹ Mathematics Department, Box 1917 Brown University, Providence, RI 02912, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Joseph H. Silverman. Arithmetic and Dynamical Degrees on Abelian Varieties. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 151-167. doi : 10.5802/jtnb.973. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.973/

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